【Quant绿皮书】Q1 Screwy pirates

Five pirates looted a chest full of 100 gold coins. Being a bunch of democratic pirates,they agree on the following method to divide the loot:The most senior pirate will propose a distribution of the coins. All pirates, including themost senior pirate, will then vote. If at least 50% of the pirates (3 pirates in this case)accept the proposal, the gold is divided as proposed. If not, the most senior pirate will befed to shark and the process starts over with the next most senior pirate., The process isrepeated until a plan is approved. You can assume that all pirates are perfectly rational:they want to stay alive first and to get as much gold as possible second. Finally, beingblood-thirsty pirates, they want to have fewer pirates on the boat if given a choicebetween otherwise equal outcomes.
How will the gold coins be divided in the end?

五名海盗抢夺了一个装满 100 枚金币的箱子。由于他们是一群民主的海盗，所以他们达成了以下的分配方法：最资深的海盗会提出金币的分配方案。所有海盗，包括最资深的海盗在内，都会进行投票。如果至少 50%的海盗（在这种情况中是 3 名海盗）接受这个提议，那么金币就会按照提议的方式进行分配。如果不接受，最资深的海盗就会被喂鲨鱼，然后分配过程会从下一个最资深的海盗开始。这个过程会一直重复，直到一个方案获得批准。可以假设所有海盗都是完全理性的：他们首先想要活下去，其次是要尽可能多地获得金币。最后，由于这些海盗是嗜血的，如果在同等结果的情况下可以选择，他们会选择让船上剩下的海盗更少。
最终金币会如何分配呢？

题目：海盗分金
类型：脑筋急转弯
思路：
由于海盗的偏好是严格有序的，使得每个子博弈都有唯一均衡；而 𝑛−1 人局面的均衡已经递归地包含了所有更小规模的信息，因此 𝑛 人局面的决策只依赖于 𝑛−1，并通过逆向归纳形成唯一且收敛的结果。
逆向推理：(资历排序A、B、C、D、E)

n	A	B	C	D	E
1					100
2				100	0
3			99	0	1
4		99	0	1	0
5	98	0	1	0	1